Rank of a matrix of S call the greatest of orders of its minors other than zero. Minors are determinants of a square matrix which turns out from initial by the choice of any lines and columns. Rg S rank is designated, and its calculation can be executed by means of carrying out elementary transformations over the set matrix or method of bordering of its minors.
Instruction
1. Write down the set matrix of S and define its greatest order. If the quantity of columns m of a matrix less than 4, makes sense to find a matrix rank by means of definition of its minors. According to definition, the rank will be equal to the biggest minor other than zero.
2. A minor of 1 order of an initial matrix is any its element. If at least one of them is other than zero (that is the matrix is not zero), it is necessary to pass to consideration of minors of the following order.
3. Calculate minors 2 orders of a matrix, consistently choosing from initial 2 lines and 2 columns. Write down the received square matrix 2х2 and calculate its determinant on formula D = a11*a22 – a12*a21 where aij – elements of the chosen matrix. If D=0, calculate the following minor, having chosen other matrix 2х2 of lines and columns as initial. Continue to consider the same way all minors 2 orders until the nonzero determinant meets. In this case you pass 3 orders to finding of minors. If all considered minors 2 orders are equal to zero, search of a rank comes to the end. The rank of a matrix of Rg S will be equal to the last order of a nonzero minor, that is in this case Rg S = 1.
4. Calculate minors 3 orders for an initial matrix, choosing already 3 lines and 3 columns for calculation of determinant of a square matrix. The determinant D of a matrix 3х3 is by the rule of a triangle of D = c11 * c22*c33 + c13 * c21*c32 + c12 * c23*c31 - c21 * c12*c33 - c13 * c22*c31 - c11 * c32*c23 where cij – elements of the chosen matrix. The same way at D=0 calculate other minors 3х3, at least one nonzero determinant will not meet yet. If all found determinants are equal to zero, the matrix rank in this case is equal 2 (Rg S = 2), that is to an order of the previous nonzero minor. When determining D, other than zero, you pass 4 orders to consideration of minors of the following. If at a certain stage the limit order of m of an initial matrix is reached, therefore, its rank is equal to this order: Rg S = m.